Extremal Area of Polygons, sliding along a Circle

Abstract

We determine all critical confiurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. In the even dimensional case non-isolated singularities occur as 'zigzag trains'.